ON DAVIS-JANUSZKIEWICZ HOMOTOPY TYPES I; FORMALITY AND RATIONALISATION

DIETRICH NOTBOHM AND NIGEL RAY

Abstract. For an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction (here called c(K)), which they showed to be homotopy equivalent to Davis and Januszkiewicz’s examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we extend work of Franz by showing that the singular cochain algebra of c(K) is formal as a differential graded noncommutative algebra. We specialise to the rationals by proving the corresponding result for Sullivan’s commutative cochain algebra, and deduce that the rationalisation of c(K) is unique for a special family of complexes K. In a sequel, we will consider the uniqueness of c(K) at each prime separately, and apply Sullivan’s arithmetic square to produce global results for this family.

1. Introduction

Over the last decade, work of Davis and Januszkiewicz [7] has popularised homotopy theoretical aspects of toric geometry amongst algebraic topologists. The results of [7] have been surveyed by Buchstaber and Panov in [4], where several further applications were developed. Their constructions have led us to consider the uniqueness of certain associated homotopy types, and our aim is to begin that study here; we focus on general issues of formality, and deduce that the rationalisations of a family of special cases are unique. In a sequel [19], we discuss the problem prime by prime, and obtain global results for members of the family by appeal to Sullivan’s arithmetic square. The general problem of uniqueness appears to be of considerable difficulty.

We work over an arbitrary commutative ring R with identity, and consider a universal set V of vertices v_{1}, …, v_{m}, ordered by their subscripts. The vertices masquerade as algebraically independent variables, which generate a graded polynomial algebra S_{R}(V) over R. The grading is defined by assigning each of the generators a common dimension, which we usually take to be 2. A function M: V → ℕ is known as a multiset on V, with cardinality |M|:= ∑_{j} M(v_{j}); it may be represented by the monomial v_{M} := ∏_{V} v^{M(v)}, or by the n-tuple of constituent vertices (v_{j_{1}}, …, v_{j_{n}}), where j_{1} ≤ ⋯ ≤ j_{n} and n = |M|. So S_{R}(V) is generated additively by the v_{M}, and v_{M} is squarefree precisely when M is a genuine subset.

Key words and phrases. Formality, Davis-Januszkiewicz spaces, homotopy colimit, model category, rationalisation, Stanley-Reisner algebra.

A simplicial complex K on V consists of a finite set of faces σ ⊆ V, closed with respect to the formation of subsets ρ ⊆ σ. Alternatively, we may interpret K as the set of squarefree monomials v_{σ} := ∏_{σ} v, which is closed under factorisation. Every simplicial complex generates a simplicial set K_{•}; for each n ≥ 0, the n-simplices K_{n} contain all M of cardinality n + 1 whose support is a face of K. The face and degeneracy operators delete and repeat the appropriate vertices respectively.

The Stanley-Reisner algebra R[K] (otherwise known as the face ring of K) is an important combinatorial invariant. It is defined as the quotient

(1.1)  S_{R}(V)/(v_{U} : U ∉ K),

and is therefore generated additively by the simplices of K_{•}. The algebraic properties of R[K] encode a host of combinatorial features of K, and are discussed in detail by Bruns and Herzog [3] and Stanley [23], for example. If K is the simplex on V, then R[K] is the polynomial algebra S_{R}(V).

For each K, Davis and Januszkiewicz defined the notion of a toric space over the cone on the barycentric subdivision of K, and showed that the cohomology of such a space is related to R[K]. The relationship follows from their application of the Borel construction, which creates a family of spaces whose cohomology ring (with coefficients in R) is isomorphic to the Stanley-Reisner algebra. All members of the family are homotopy equivalent to a certain universal example, and we refer to any space which shares their common homotopy type as a Davis-Januszkiewicz space. The isomorphisms equip R[K] with a natural grading, which agrees with that induced from S_{R}(V). Subsequently, Buchstaber and Panov [4] defined a CW-complex whose cohomology ring is also isomorphic to R[K]. They confirmed that their complex is a Davis-Januszkiewicz space by giving an explicit homotopy equivalence with the universal example. In [21], their space is described as the pointed colimit colim^{+}(B^{K}) of a certain cat(K)-diagram B^{K}, which assigns the cartesian product B^{σ} to each face σ of K. Here B denotes the classifying space of the circle, or CP^{∞}, and cat(K) is the category of faces and inclusions.

We say that a space X realises the Stanley-Reisner algebra of K whenever there is an algebra isomorphism H^{∗}(X; R) ≅ R[K]. We denote the rationalisation of X by X_{0}, and write Aut_{ho}X < End_{ho}X for the homotopy classes of self-equivalences of X, considered as a subgroup of the homotopy classes of self-maps with respect to composition.

The contents of each section are as follows.

In Section 2 we describe our notation and prerequisites, including those aspects of model category theory which provide a useful context for exponential diagrams and their cohomology. We also explain why it is equally acceptable to work with the unpointed colimit c(K) := colim(B^{K}). We introduce the Stanley-Reisner algebra in Section 3, and show that the Bousfield-Kan spectral sequence for H^{∗}(c(K); R) collapses by analysing higher limits of certain cat(K)-diagrams. In Section 4 we apply similar techniques to prove the formality of the singular cochain algebra C^{∗}(c(K); R). Finally, we specialise to the case R = ℚ in Section 5, where we confirm that Sullivan’s commutative cochain algebra A_{PL}(c(K)) is formal in the commutative sense. We deduce that ℚ[K] determines the rationalisation c(K)_{0} whenever it is a complete intersection, and discuss the corresponding automorphism group Aut_{ho}c(K)_{0}.

We would like to develop the results of Section 4 in the model category of E_{∞} algebras, which are emerging as the integral analogue of differential graded commutative algebras over ℚ. Recent work of Mandell [17] shows that c(K) is classified by the E_{∞} algebra C^{∗}(c(K); ℤ), at least up to weak equivalence of nilpotent spaces. As we shall explain in the sequel, we have no evidence that C^{∗}(c(K); R) is always formal in this stricter sense.

The authors are especially grateful to the organisers of the International Conference on Algebraic Topology, which was held on the Island of Skye in June 2001. The Conference provided the opportunity for valuable discussion with several colleagues, amongst whom Octavian Cornea, Kathryn Hess, and Taras Panov deserve special mention. Without that remarkable and stimulating environment, our work would not have begun. We are more recently indebted to John Greenlees, who brought to our attention a fundamental misconception in a previous version of this article. We must also thank the London Mathematical Society, whose support of the Transpennine Topology Triangle has enabled our collaboration to continue and expand.

2. Background

We begin by establishing our notation and prerequisites, recalling various aspects of Davis-Januszkiewicz spaces. We refine results of [21] in the context of model category theory, referring readers to [9] and [14] for background details. Following [24], we adopt the model category top of k-spaces and continuous functions as our topological workplace. Weak equivalences induce isomorphisms in homotopy, fibrations are Serre fibrations, and cofibrations have the left lifting property with respect to acyclic fibrations. Every function space Y^{X} is endowed with the corresponding k-topology. Many of the spaces we consider have a distinguished basepoint ∗, and we write top_{+} for the model category of pairs (X, ∗) and basepoint preserving maps. We often require the inclusion of ∗ to be a closed cofibration, in which case X is said to be well-pointed; this is automatic when X is a CW-complex and ∗ its 0–skeleton.

Given a small category a, we refer to a covariant functor D: a → r as an a-diagram in r. Such diagrams are themselves the objects of a category ^{[}a,r^{]}, whose morphisms are natural transformations of functors. We may interpret any object X of r as a constant diagram, which maps every object of a to X and every morphism to the identity. We describe a as finite whenever the total collection of morphisms is finite. Example 2.1. (1) For each integer n ≥ 0, the category ord(n) has objects 0, 1, …, n, equipped with a single morphism k → m when k ≤ m. An ord(n)-diagram

(2.2)  X_{0} ⟶^{f_{1}} X_{1} ⟶^{f_{2}} ⋯ ⟶^{f_{n}} X_{n}

consists of n composable morphisms in r.

(2) The category Δ has objects (n) := {0, 1, …, n} for n ≥ 0, and morphisms the nondecreasing functions; then Δ^{op}- and Δ-diagrams are simplicial and cosimplicial objects of r respectively. In particular, Δ: Δ → top is the cosimplicial space which assigns the standard n-simplex Δ(n) to each object (n).

Given objects X_{0} and X_{1} of r, we write the set of morphisms X_{0} → X_{1} as r(X_{0}, X_{1}); when r is small, the diagrams (2.2) form a set for every n > 1. For any r it is often convenient to abbreviate ^{[}Δ^{op},r^{]} to sr, and write a generic simplicial object as D_{•}. In particular, sset denotes the category of simplicial sets Y_{•}.

From this point on we work with an abstract simplicial complex K, whose faces σ are subsets of the vertices V. We assume that the empty face belongs to K, and write K^{×} when it is expressly omitted. The integer |σ| − 1 is known as the dimension of σ, and written dim σ; its maximum value dim K is the dimension of K. When K contains every subset of V, we may call it the simplex Δ(V) on V. Each face of K therefore determines a subsimplex Δ(σ), whose boundary ∂(σ) is given by deleting the subset σ. We also require the link ℓ_{K}(σ), whose faces consist of those τ \ σ for which σ ⊆ τ in K. Definition 2.3. For any simplicial complex K, the small category cat(K) has objects the faces of K and morphisms the inclusions i_{σ,τ}: σ ⊆ τ. The empty face ∅ is an initial object, and the maximal faces μ admit only identity morphisms. The opposite category cat^{op}(K) has morphisms p_{τ,σ} := i_{σ,τ}^{op}: τ ⊇ σ, and ∅ is final.

The nondegerate simplices of the nerve N_{•}cat(K) form the cone on the barycentric subdivision K′, and those of N_{•}cat(K^{×}) correspond to the subcomplex K′. So the classifying space Bcat(K), formed by realising the nerve, is a contractible CW-complex, and Bcat(K^{×}) is a subcomplex homeomorphic to K. We shall study cat(K)- and cat^{op}(K)- diagrams D in various algebraic and topological categories r. Usually, r is pointed by an object ∗, which is both initial and final; unless stated otherwise, we then assume that D(∅) = ∗.

For each face σ, the overcategories cat(K) ↓ σ and cat(K) ⇓ σ are given by restricting attention to those objects ρ for which ρ ⊆ σ and ρ ⊂ σ respectively. The undercategories σ ↓ cat(K) and σ ⇓ cat(K) are defined by the objects σ ⊆ τ and σ ⊂ τ. It follows from the definitions that

cat(K)↓σ = cat(Δ(σ)), cat(K)⇓σ = cat(∂(σ)),
σ ↓cat(K) = cat(ℓ_{K}(σ)) and σ ⇓cat(K) = cat(ℓ_{K}(σ)^{×}).

Dimension may be interpreted as a functor dim: cat(K) → ord(m − 1), which is a linear extension in the sense of [14]; thus cat(K) is direct and cat^{op}(K) is inverse.

For any model category r, we may therefore follow Hovey and impose an associated model structure on the category of diagrams ^{[}cat(K),r^{]}. Weak equivalences e: C → D are given objectwise, in the sense that e(σ): C(σ) → D(σ) is a weak equivalence in r for every face σ of K. Fibrations are also given objectwise. To describe the cofibrations, we consider the restrictions of C and D to the overcategories cat(∂(σ)), and write L_{σ}C and L_{σ}D for their respective colimits; L_{σ} is the latching functor of [14]. Then g: C → D is a cofibration whenever the induced maps

(2.4)  C(σ) ⨿_{L_{σ}C} L_{σ}D ⟶ D(σ)

are cofibrations in r for every face σ. Alternatively, the methods of Chacholski and Scherer [6] lead to the same model structure on ^{[}cat(K),r^{]}.

There is a dual model category structure on ^{[}cat^{op}(K),r^{]}, where weak equivalences and cofibrations are given objectwise. To describe fibrations f: C → D, we consider the restrictions of C and D to the undercategories cat^{op}(∂(σ)), and write M_{σ}C and M_{σ}D for their respective limits; M_{σ} is the matching functor of [14]. Then f is a fibration whenever the induced maps

(2.5)  C(σ) ⟶ D(σ) ×_{M_{σ}D} M_{σ}C

are fibrations in r for every face σ. Definition 2.6. For any CW-pair (X, ∗), the exponential pair of diagrams (X^{K}, X_{K}) consists of functors

X^{K}: cat(K) ⟶ top_{+} and X_{K}: cat^{op}(K) ⟶ top_{+},

which assign the cartesian product X^{σ} to each face σ of K. The value of X^{K} on i_{σ,τ} is the cofibration X^{σ} → X^{τ}, where the superfluous coordinates are set to ∗, and the value of X_{K} on p_{τ,σ} is the fibration X^{τ} → X^{σ}, defined by projection. The pair are twins, in the sense that X_{K}(p′) · X^{K}(i) = X^{K}(j′) · X_{K}(q) for every pullback square

σ ∩ σ′ ⟶^{j′} σ′
j↓  ↓^{i′}
σ ⟶_{i} τ

in cat(K), where p′ = (i′)^{op} and q = j^{op}.

The properties of twin diagrams are analogous to those of a Mackey functor. They include, for example, the fact that each X^{K}(i) has left inverse X_{K}(p), where p = i^{op}. Our applications in Theorem 3.10 are reminiscent of [15], where the acyclicity of certain Mackey functors is established.

The colimit colim X^{K} is a subcomplex of X^{V}, whose inclusion r_{K} is induced by interpreting the elements σ of K as faces of the (m − 1)-simplex Δ(V). Composing r_{K} with any of the natural maps X^{σ} → colim X^{K} yields the standard inclusion X^{σ} → X^{V}. We note that colim X^{K} is pointed by X^{∅}, otherwise known as the basepoint ∗, and is homeomorphic to the pointed colimit colim^{+} X^{K} of [21].

We wish to study homotopy theoretic properties of colim X^{K} in favourable cases. Yet the colimit functor behaves particularly poorly in this context, because objectwise equivalent diagrams may well have homotopy inequivalent colimits. The standard procedure for dealing with this situation is to introduce the left derived functor, known as the homotopy colimit. Following [13], for example, hocolim X^{K} may be described by the two-sided bar construction B(∗, cat(K), X^{K}) in top. We note that hocolim X^{K} is also pointed, and is related to the pointed homotopy colimit hocolim^{+} X^{K} of [21] by the cofibre sequence

Bcat(K) ⟶ hocolim X^{K} ⟶^{f} hocolim^{+} X^{K}

of [2]. Since Bcat(K) is contractible, f is a weak equivalence. We may therefore concentrate on hocolim X^{K}, and so avoid basepoint complications when working with function spaces in [19]. Lemma 2.7. Every exponential diagram is cofibrant in ^{[}cat(K),top^{]}.

Proof. The initial cat(K)-diagram in top is the constant diagram ∗, so X^{K} is cofibrant whenever the inclusion ∗ → X^{K} is a cofibration. By (2.4), it suffices to show that the map X^{∂(σ)} → X^{σ} is a cofibration for every face σ of K. But this map includes the fat wedge in the cartesian product, and the result follows. □

An immediate consequence of Lemma 2.7 is that the natural projection

(2.8)  hocolim X^{K} ⟶ colim X^{K}

is a homotopy equivalence. This illustrates one of the fundamental results of model category theory, and is sometimes called the Projection Lemma [25].

3. Integral Cohomology and Limits

In this section we work in the category mod_{R} of R-modules, and study the cohomology of exponential diagrams B^{K}, where B is the classifying space of the circle. For this case only, we abbreviate (2.8) to hc(K) → c(K). We focus on the relationship between the Stanley-Reisner algebra R[K] and the Bousfield-Kan spectral sequence for H^{∗}(hc(K); R).

We begin by investigating the cohomology of c(K). To simplify applications in later sections, we consider an arbitrary pair of twin diagrams (D_{K}, D^{K}),

(3.1)  D_{K}: cat(K) ⟶ mod_{R} and D^{K}: cat^{op}(K) ⟶ mod_{R}.

Thus D^{K}(p′) · D_{K}(i) = D_{K}(j) · D^{K}(q′) for every pullback square i · j = i′ · j′ in cat(K). In particular, D^{K}(p) has right inverse D_{K}(i) for every morphism p = i^{op}. Such pairs arise, for example, from any contravariant functor D: top → mod_{R}, by composing with the exponential twins of Definition 2.6. So (D_{K}, D^{K}) = (D · B_{K}, D · B^{K}), and functoriality ensures the diagrams are twins. In this case we may apply D to the natural maps B^{σ} → c(K) ⟶^{r_{K}} B^{V}, and obtain homomorphisms

(3.2)  D(B^{V}) ⟶^{D(r_{K})} D(c(K)) ⟶^{h_{K}} lim D^{K}

in mod_{R}.

By way of example, we consider the case D = H^{2j}(−, R), for any j ≥ 0. For every face σ of K, the space B^{σ} is an Eilenberg-Mac Lane space H(ℤ^{σ}, 2), and may be expressed as the realisation of a simplicial abelian group H_{•}(ℤ^{σ}, 2) whenever convenient [18]. As a CW-complex, the cells of B^{σ} are concentrated in even dimensions, and correspond to the simplices v_{M} of Δ(σ)_{•}. The cellular cohomology group H^{2j}(B^{σ}; R) is therefore isomorphic to the free R-module generated by those v_{M} for which |M| = j and the support of M is a subset of σ. The diagram D^{K} of (3.1) becomes

(3.3)  H^{2j}(B^{K}; R): cat^{op}(K) ⟶ mod_{R},

whose value on p_{τ,σ} is the homomorphism which fixes v_{M} whenever the support of M lies in σ, and annihilates it otherwise; the right inverse is the inclusion induced by D_{K}. When D = H^{2j+1}(−, R), the diagram is zero.

In the case of cohomology, we may combine the diagrams (3.3) into a graded version

(3.4)  H^{∗}(B^{K}; R): cat^{op}(K) ⟶ gmod_{R}^{+},

taking values in the category of augmented graded R-modules. The cup product on each of the constituent submodules H^{∗}(B^{σ}; R) is given by the product of monomials x_{L}x_{M} = x_{L+M}, as follows from the case of a single vertex. In other words, the cohomology ring H^{∗}(B^{σ}; R) is isomorphic to the polynomial algebra S_{R}(σ). So H^{∗}(B^{K}; R) actually takes values in the category gca_{R}^{+} of augmented graded commutative R-algebras, and maps the morphism p_{τ,σ} to the projection S_{R}(τ) → S_{R}(σ); the right inverse is again inclusion.

The homomorphisms (3.2) may similarly be combined as

(3.5)  S_{R}(V) ⟶^{r_{K}^{∗}} H^{∗}(c(K); R) ⟶^{h_{K}} lim H^{∗}(B^{K}; R),

where the limit is taken in gmod_{R}^{+}. Since (3.4) is a diagram of algebras, the limit inherits a multiplicative structure, and it is equally appropriate to interpret (3.5) in gca_{R}^{+}. The composition h_{K} · r_{K}^{∗} is induced by the projections S_{R}(V) → S_{R}(σ). In this case, we make one further observation. Proposition 3.6. The homomorphism r_{K}^{∗} is epic, and its kernel is the ideal (v_{U} : U ∉ K).

Proof. In each dimension 2j, the cells v_{M} of B^{V} correspond to the multisets on V with |M| = j. The cells of c(K) form a subset, given by those M whose support is a face of K. Hence r_{K}^{∗} is epic, and its kernel is generated by the remaining cells. These coincide with the 2j–dimensional additive generators of the ideal (v_{U} : U ∉ K). □

So there is an isomorphism R[K] ≅ H^{∗}(c(K); R) of the Stanley-Reisner algebra (1.1), which plays a central rôle in [4].

Returning to our study of the twins (D_{K}, D^{K}), the following definition identifies a further important property. Definition 3.7. A diagram F^{K}: cat^{op}(K) → mod_{R} of R-modules (graded or otherwise) is fat if the natural map F^{K}(σ) → lim F^{∂(σ)} is an epimorphism for every face σ of K.

The terminology acknowledges the relationship between ∂(σ) and the fat wedge described in Lemma 2.7. Lemma 3.8. The twin D^{K} is fat.

Proof. We consider an arbitrary face ρ of K, whose vertices we label w_{k} for 0 ≤ k ≤ d; thus d = dim ρ. We write μ_{k} := ρ \ w_{k} for the maximal faces of ∂(ρ), and abbreviate the morphism p_{ρ,μ_{k}} to p_{k} for 0 ≤ k ≤ d.

The definition of lim ensures that L:= lim D^{∂(ρ)} appears in an exact sequence

0 ⟶ L ⟶ ∏_{ρ⊃σ} D^{K}(σ) ⟶^{δ} ∏_{τ⊃σ} D^{K}(σ),

where δ(u)(τ ⊃ σ) = u(σ) − D^{K}(p_{τ,σ})u(τ) for any u ∈ ∏_{σ⊂ρ} D^{K}(σ). Hence u ∈ L is determined by the values u(μ_{k}). The natural projection D^{K}(ρ) → ∏_{ρ⊃σ} D^{K}(σ) therefore factors through L, and it remains for us to find u(ρ) ∈ D^{K}(ρ) such that D^{K}(p_{k})(u(ρ)) = u(μ_{k}) for every 0 ≤ k ≤ d.

We set u(ρ):=∑_{ρ⊃σ}(−1)^{|ρ\σ|+1}D_{K}(i_{σ,ρ})u(σ). The fact that D_{K} and D^{K} are twins implies that

D^{K}(p_{k})D_{K}(i_{σ,ρ})u(σ) = \begin{cases}D_{K}(i_{σ\w_{k},μ_{k}})u(σ\w_{k}) & if w_{k} ∈ σ \\ D_{K}(i_{σ,μ_{k}})u(σ) & otherwise, \end{cases}

for every 0 ≤ k ≤ d; thus D^{K}(p_{k})u(ρ) is given by

∑_{σ∌w_{k}}(−1)^{|ρ\σ|+1}D_{K}(i_{σ,μ_{k}})u(σ) + ∑_{τ∋w_{k}}(−1)^{|ρ\τ|+1}D_{K}(i_{τ\w_{k},μ_{k}})u(ρ\w_{k}).

But we may write u(σ) as u((σ ∪ w_{k})\w_{k}) for any σ ∌ w_{k} other than μ_{k}. So the summands cancel in pairs, leaving u(μ_{k}) as required. □

For cohomology, Lemma 3.8 contributes to our analysis of c(K). The homotopy equivalence (2.8) provides a cohomology decomposition [8], in the sense that the cohomology algebra H^{∗}(c(K); R) may be computed by the Bousfield-Kan spectral sequence [2]

E_{2}^{i,j} ⟹ H^{i+j}(hc(K); R),

where E_{2}^{i,j} is isomorphic to the ith derived functor lim^{i} H^{j}(B^{K}; R) for every i, j ≥ 0. The vertical edge homomorphism coincides with the map h_{K} of (3.5). Lemma 3.8 is required for our computation of these limits, and Corollary 3.12 will confirm that the cohomology decomposition is sharp in Dwyer’s language. Our proof uses the calculus of functors and their limits; the appropriate prerequisites may be deduced from Gabriel and Zisman [12, Appendix II §3], by dualising their results for colimits.

In particular, we follow [12] (as expounded in [20], for example) by calculating lim^{i} D^{K} as the i th cohomology group of a certain cochain complex (C^{∗}(D^{K}), δ) of R-modules. The groups are defined by

C^{n}(D^{K}) := ∏_{σ_{0}⊇...⊇σ_{n}} D^{K}(σ_{n}) for n ≥ 0 ,

and the differential δ := ∑_{k=0}^{n}(−1)^{k}δ^{k} is defined on u ∈ C^{n}(D^{K}) by

δ^{k}(u)(σ_{0} ⊇ ⋯ ⊇ σ_{n+1}) := \begin{cases}u(σ_{0} ⊇ ⋯ ⊇ σ̂_{k} ⊇ ⋯ ⊇ σ_{n+1}) & for k ≠ n + 1 \\ D^{K}(p_{σ_{n},σ_{n+1}})u(σ_{0} ⊇ ⋯ ⊇ σ_{n}) & for k = n + 1. \end{cases}

We may replace C^{∗}(D^{K}) by its quotient N^{∗}(D^{K}) of normalised cochains, for which the faces σ_{0} ⊃ ⋯ ⊃ σ_{n} are required to be distinct. Lemma 3.9. Given a diagram D: cat^{op}(K) → mod_{R}, and a maximal face μ of K, then

lim^{i}D = \begin{cases}D(μ) & for i = 0 \\ 0 & for i > 0 \end{cases}

whenever D(σ) = 0 for all σ ≠ μ.

Proof. Since μ is maximal, the only morphism σ ⊇ μ is the identity. So the normalised chain complex N^{∗}(D) is D(μ) in dimension 0, and 0 in higher dimensions, as required. □

Theorem 3.10. For any fat diagram F^{K}: cat^{op}(K) → mod_{R}, we have that lim^{i} F^{K} = 0 for all i > 0; in particular, lim^{i} D^{K} = 0 for every twin D^{K}.

Proof. We proceed by induction on the total number of faces f(K); the result obviously holds for the initial example K = ∅, where f(K) = 0. Our inductive hypothesis is that lim^{i} F^{K} vanishes whenever K satisfies f(K) ≤ f.

We therefore consider an arbitrary complex K with f(K) = f + 1, and write J ⊂ K for the subcomplex obtained by deleting a single maximal face μ. The inclusion of J defines a functor G: cat^{op}(J) → cat^{op}(K), whose induced functor G^{∗}: ^{[}cat^{op}(K),mod_{R}^{]} → [cat^{op}(J),mod_{R}^{]} acts by restriction, and admits a right adjoint G_{∗}, known as the right Kan extension [16]. In particular, G_{∗}F^{J} is given on σ ∈ K by lim F^{∂(μ)} when σ = μ, and F^{σ} otherwise.

But F^{μ} → lim D^{∂(μ)} is an epimorphism, by Lemma 3.8, so the natural transformation F^{K} → G_{∗}F^{J} is epic on every face of K, and its kernel H is zero on every face except μ. We acquire a short exact sequence of functors

0 ⟶ H ⟶ F^{K} ⟶ G_{∗}F^{J} ⟶ 0,

which induces a long exact sequence of higher limits. By Lemma 3.9, this collapses to a sequence of isomorphisms

(3.11)  lim^{i}F^{K} ≅ lim^{i}G_{∗}F^{J},

for i ≥ 1. We now apply the composition of functors spectral sequence [5], [12]

lim^{n}G_{∗}^{i}F^{J} ⟹ lim^{n+i}F^{J}.

Here G_{∗}^{i} denotes the ith derived functor of G_{∗}; it may be evaluated on any face σ of K as lim^{i} F^{∂(σ)}, and therefore vanishes for i > 0, by inductive hypothesis. So the spectral sequence collapses onto the first row of the E^{2} page, from which we obtain isomorphisms lim^{n} G_{∗}F^{J} ≅ lim^{n} F^{J} for all n ≥ 0. Since the inductive hypothesis applies to J, we deduce that lim^{n} G_{∗}F^{J} = 0 for every n > 0. Combining this with (3.11) concludes the proof. □

Corollary 3.12. The Bousfield-Kan spectral sequence for B^{K} collapses at the E_{2} page; it is concentrated along the vertical axis, and given by

lim^{i}H^{j}(B^{K}; R) = \begin{cases}lim H^{j}(B^{K}; R) & if i = 0 \\ 0 & otherwise. \end{cases}

Proof. The result follows immediately from Theorem 3.10 by letting D^{K} be H^{j}(−; R) for every j ≥ 0. □

Corollary 3.12 confirms that the edge homomorphism h_{K} is an isomorphism in gca_{R}^{+}. When combined with (3.5) and Proposition 3.6 it implies that the natural map

R[K] ≅ H^{∗}(c(K); R) ⟶^{h} lim H^{∗}(B^{K}),

which is induced by the projections R[K] → S_{R}(σ), is also an isomorphism. This may be proven directly, by refining the methods of Proposition 3.6.

4. Integral Formality

In this section we study the formality of c(K) over our arbitrary commutative ring R, and construct a zig-zag of quasi-isomorphisms between the singular cochain algebra C^{∗}(c(K); R) and its cohomology ring.

We work in the model category dga_{R}^{+} of differential graded R-algebras with augmentation. The model structure arises by interpreting dga_{R}^{+} as the category of monoids in the monoidal model category dgmod_{R} of unbounded cochain complexes over R. The latter is isomorphic to Hovey’s category of unbounded chain complexes [14], and the model structure is induced on dga_{R}^{+} by checking that it satisfies the monoid axiom of [22]. As Schwede and Shipley confirm, weak equivalences are zig-zags of quasi-isomorphisms and fibrations are epimorphisms. Cofibrations are defined by the appropriate lifting property, and are necessarily degreewise split injections. We emphasise that the objects of dga_{R}^{+} need not be commutative.

A differential graded R-algebra C^{∗} is formal in dga_{R}^{+} whenever there is a zig-zag of quasi-isomorphisms

(4.1)  H(C^{∗}) ⟶^{∼} ⋯ ⟵^{∼} C^{∗},

where we follow the convention of assigning the zero differential to the cohomology algebra H(C^{∗}). Our aim is to show that the cochain algebra C^{∗}(c(K); R) is always formal in dga_{R}^{+}. This extends Franz’s result [11], which only applies to complexes arising from smooth fans.

We begin by choosing D to be the j-dimensional cochain functor C^{j}(−; R) in (3.1), thus creating twin diagrams (C^{j}(B_{K}; R), C^{j}(B^{K}; R)) for each j ≥ 0. As in (3.4), we may consider the graded version C^{∗}(−; R) in dgmod_{R}. In fact its values are always R-algebras, with respect to the cup product of cochains. The product is not commutative, but the procedure for forming the limit of a dga_{R}^{+}-diagram remains the same; work in dgmod_{R}^{+}, and superimpose the induced multiplicative structure.

For the Eilenberg-Mac Lane space B = H(ℤ, 2), we let v denote a generator of H^{2}(B; R), which is isomorphic to R. We choose a cocycle ψ_{v} representing v in C^{2}(B; R), and define a homomorphism ψ: H^{∗}(B; R) → C^{∗}(B; R) by ψ(v^{k}) = ψ_{v}^{k}, for all k ≥ 0. By construction, ψ is multiplicative, and is a quasi-isomorphism in dga_{R}^{+}. We extend this procedure to H^{∗}(B^{V}; R) via the zig-zag

(4.2)  H^{∗}(B^{V}; R) ⟶^{≅} H^{∗}(B; R)^{⊗V} ⟶^{ψ^{⊗}} C^{∗}(B; R)^{⊗V} ⟵^{ez} C^{∗}(B^{V}; R)

of quasi-isomorphisms in dga_{R}^{+}; the leftmost map is the Künneth isomorphism, and ez arises from the dual of the Eilenberg-Zilber map as the composite

Hom(C_{∗}(B^{V}); R) ⟶ Hom(C_{∗}(B)^{⊗V}; R) ≅ Hom(C_{∗}(B); R)^{⊗V} .

Of course (4.2) confirms that C^{∗}(B^{n}; R) is formal in dga_{R}^{+}, for every n ≥ 1.

For each B^{σ}, we may project (4.2) onto the corresponding zig-zag of quasi-isomorphisms. The results are compatible by naturality, and combine to give morphisms

H^{∗}(B^{K}; R) ⟶^{≅} H^{∗}(B; R)^{⊗K} ⟶^{ψ^{⊗}} C^{∗}(B; R)^{⊗K} ⟵^{ez} C^{∗}(B^{K}; R)

of cat^{op}(K)-diagrams. Taking limits in dga_{R}^{+} yields homomorphisms

lim H^{∗}(B^{K}; R) ⟶^{≅} lim H^{∗}(B; R)^{⊗K} ⟶^{ψ^{⊗}} lim C^{∗}(B; R)^{⊗K} ⟵^{ez} lim C^{∗}(B^{K}; R).

Lemma 4.3. The leftmost homomorphism is an isomorphism in dga_{R}^{+}, and the remaining two are quasi-isomorphisms.

Proof. The limit of the Künneth isomorphisms is automatically an isomorphism.

A diagram D^{K}: cat^{op}(K) → dga_{R}^{+} is fibrant whenever the projection onto the constant diagram 0 is a fibration. By (2.5) this occurs precisely when D^{K} is fat, and therefore holds for H^{∗}(B^{K}; R) and C^{∗}(B^{K}; R) by Lemma 3.8; it follows for H^{∗}(B; R)^{⊗K} by the Künneth isomorphism. So far as C^{∗}(B; R)^{⊗K} is concerned, we note that singular cochains determine a pair of diagram twins (C_{⊗K}, C^{⊗K}) in dga_{R}^{+}. Both functors assign C^{∗}(B; R)^{⊗σ} to the face σ. The value of C_{⊗K} on i_{σ,τ} is the inclusion C^{∗}(B; R)^{⊗σ} → C^{∗}(B; R)^{⊗τ}, and the value of C^{⊗K} on p_{τ,σ} is the projection C^{∗}(B; R)^{⊗τ} → C^{∗}(B; R)^{⊗σ}; these require the augmentation. Hence C^{⊗K} is also fat, and C^{∗}(B; R)^{⊗K} is fibrant. So the remaining two homomorphisms are objectwise equivalences of fibrant diagrams, and therefore induces weak equivalences of limits by [14]. □

Lemma 4.4. The natural homomorphism g: C^{∗}(c(K); R) → lim C^{∗}(B^{K}; R) is a quasiisomorphism in dga_{R}^{+}.

Proof. The edge isomorphism h_{K} of Corollary 3.12 factorises as

H(C^{∗}(c(K); R)) ⟶^{H(g)} H(lim C^{∗}(B^{K}; R)) ⟶^{l} lim H^{∗}(B^{K}; R)

in dga_{R}^{+}, where l is induced by the collection of compatible homomorphisms

H(lim C^{∗}(B^{K}; R)) → H^{∗}(B^{σ}; R).

Now let d be the differential on C^{j}(−; R) for every j ≥ 0, and define the cycle and boundary functors Z^{j}, I^{j}: top → mod_{R} as the kernel and image of d respectively. They determine diagram twins, and therefore fat functors Z^{K}, I^{K}: cat^{op}(K) → mod_{R}. So Theorem 3.10 confirms that lim^{i} Z^{j}(B^{K}; R) = lim^{i} I^{j}(B^{K}; R) = 0 for all i > 0 and j ≥ 0. It follows immediately that l is an isomorphism, and therefore that H(g) is an isomorphism, as sought. □

We may now complete our analysis of C^{∗}(c(K); R). Theorem 4.5. The differential graded R-algebras C^{∗}(c(K); R) is formal in dga_{R}^{+}.

Proof. Combining Corollary 3.12 with Lemmas 4.3 and 4.4 yields a zig-zag

H^{∗}(c(K); R) ⟶^{h} lim H^{∗}(B^{K}; R) ⟶ … ⟵ lim C^{∗}(B^{K}; R) ⟵^{g} C^{∗}(c(K); R)

of quasi-isomorphisms, as required by (4.1). □

Remark 4.6. The proof of Theorem 4.5 extends to exponential diagrams X^{K} for which C^{∗}(X; R) is formal in dga_{R}^{+} and the Künneth isomorphism H^{∗}(X^{V}; R) ≅ H^{∗}(X; R)^{⊗V} holds. We replace ψ in (4.2) by the corresponding zig-zag H^{∗}(X; R) ⟶^{∼} ⋯ ⟵^{∼} C^{∗}(X; R) of quasi-isomorphisms, and repeat the remainder of the argument above.

5. Rational Formality

In our final section we turn to the rational case R = ℚ, and confirm the formality of Sullivan’s algebra of rational cochains on c(K) in the commutative setting. This involves stricter conditions than those for general R, and has deeper topological consequences. In particular, it leads us to a minimal model whenever ℚ[K] is a complete intersection ring, and thence to rational uniqueness. In other words, for any complete intersection ℚ[K], the existence of an isomorphism H^{∗}(X; Q) ≅ H^{∗}(c(K); Q) implies that there is a rational homotopy equivalence X ≃ c(K), at least for nilpotent spaces X. We refer readers to Bousfield and Gugenheim [1] for details of the model category of differential graded commutative ℚ-algebras, and to Félix, Halperin and Thomas [10] for background information on rational homotopy theory.

We begin by replacing C^{∗}(X; R) with Sullivan’s rational algebra A_{PL}(X) of polynomial forms[10]. The commutativity of the latter is crucial, and suggests we work in the category dgca_{ℚ}^{+} of differential graded commutative ℚ-algebras [1]. The existence of a model structure is assured by working over a field; as before, weak equivalences are zig-zags of quasi-isomorphisms, fibrations are epimorphisms, and cofibrations are defined by the appropriate lifting properties.

For each s ≥ 0, we write the differential algebra of rational polynomial forms on the standard s-simplex as ∇_{s}(∗). It is an object of dgca_{ℚ}^{+}. For each t ≥ 0, the forms of dimension t define a simplicial vector space ∇_{•}(t) over ℚ, and ∇_{•}(∗) becomes a simplicial object in dgca_{ℚ}^{+}. So

A^{∗}(Y_{•}) := sset(Y_{•}, ∇_{•}(∗))

is also an object of dgca_{ℚ}^{+}, which is weakly equivalent to the normalised cochain complex N^{∗}(Y_{•}; ℚ). Then A_{PL}(X) is defined as A^{∗}(S_{•}X), where S_{•} denotes the total singular complex functor top → sset. The PL de Rham Theorem [1] asserts that the cohomology algebra H(A_{PL}(X)) is naturally isomorphic to H^{∗}(X; ℚ). As usual, we invest cohomology algebras with the zero differential.

A differential graded commutative ℚ-algebras A^{∗} is formal in dgca_{ℚ}^{+} whenever there is a zig-zag of quasi-isomorphisms

(5.1)  H(A^{∗}) ⟶^{∼} ⋯ ⟵^{∼} A^{∗}

in dgca_{ℚ}^{+}. A topological space X is rationally formal whenever A_{PL}(X)) satisfies (5.1).

For any such X, a minimal Sullivan model may be constructed directly from the algebra H^{∗}(X; ℚ). One of the fundamental principles of rational homotopy theory asserts that nilpotent rational homotopy types correspond bijectively to isomorphism classes of minimal ℚ-algebras, at least when both have finite type. Our remaining goal is therefore to show that c(K) is rationally formal, and to consider the implications for the uniqueness of spaces X realising ℚ[K]. The proof parallels that for general R, but the need to respect commutativity forces several changes of detail.

We choose D to be A_{PL} in (3.1), creating diagram twins (A_{PL}(B_{K}), A_{PL}(B^{K})) in dgca_{ℚ}^{+}. As before, we form limits by working in dgmod_{Q}^{+}, and superimposing the induced multiplicative structure. Applying cohomology yields the twins (H^{∗}(B_{K}; ℚ), H^{∗}(B^{K}; ℚ)), whose value on each face σ is S_{ℚ}(σ) in dgmod_{Q}^{+}. Both A_{PL}(B^{K}) and H^{∗}(B^{K}; ℚ) are fat, by Lemma 3.8.

Using the fact that H(A_{PL}(B^{V})) is isomorphic to S_{ℚ}(V), we choose cocycles ϕ_{j} in A_{PL}(B^{V}) representing v_{j} for every 1 ≤ j ≤ m. We may then define a homomorphism

(5.2)  ϕ: H^{∗}(B^{V}; ℚ) ⟶ A_{PL}(B^{V})

by ϕ(v_{j}) = ϕ_{j}, because A_{PL}(B^{V}) is commutative. Moreover, ϕ is a quasi-isomorphism, reflecting the rational formality of the Eilenberg-Mac Lane space H(ℚ^{V}; 2). By restriction, we interpret the ϕ_{j} as cocycles in A_{PL}(B^{σ}) for every face σ. They then represent v_{j} in S_{ℚ}(σ) when σ contains v_{j}, and 0 otherwise. We obtain compatible quasi-isomorphisms on each B^{σ}, which combine to create a map

ϕ: H^{∗}(B^{K}; ℚ) ⟶ A_{PL}(B^{K})

of cat^{op}(K)-diagrams in dgca_{ℚ}^{+}. It is an objectwise weak equivalence. Taking limits yields a homomorphism

l(ϕ): lim H^{∗}(B^{K}; ℚ) ⟶ lim A_{PL}(B^{K})

of differential graded commutative algebras over ℚ. Lemma 5.3. The homomorphism l(ϕ) is a quasi-isomorphism in dgca_{ℚ}^{+}.

Proof. Both diagrams are fat, and therefore fibrant by (2.5). So ϕ induces a weak equivalence of limits. □

Because the B^{σ} are Eilenberg-Mac Lane spaces, it is convenient to complete our proof of Theorem 5.5 in terms of simplicial sets. We may then take advantage of the fact that A^{∗} converts colimits in sset to limits in dgca_{ℚ}^{+}, for reasons which are purely set-theoretic.

We denote the realisation functor sset → top by |−|. Given an arbitrary simplicial set Y_{•}, we have

(5.4)  A_{PL}(|Y_{•}|) := A^{∗}(S_{•}|Y_{•}|) ≅ A^{∗}(Y_{•}),

where the second isomorphism is induced by the natural equivalence Y_{•} → S_{•}|Y_{•}|). For each face σ of K, we choose |H_{•}(ℤ^{σ}; 2)| as our model for B^{σ}; it is well-pointed, by the cofibration induced by the inclusion of the trivial subgroup {0} → ℤ^{σ}. We write H_{•}^{K} for the corresponding diagram of simplicial sets, which takes the value H_{•}(ℚ^{σ}; 2) on σ. Theorem 5.5. The space c(K) is rationally formal.

Proof. Since realisation is left adjoint to S_{•}, it commutes with colimits. So we may write

c(K) = colim |H_{•}^{K}| ≅ | colim H_{•}^{K}|,

where the second colimit is taken in sset. Applying (5.4) yields

lim A_{PL}(B^{K}) ≅ lim A^{∗}(H_{•}^{K}) ≅ A^{∗}(colim H_{•}^{K}) ≅ A_{PL}(c(K)),

where the first and third isomorphisms define A_{PL}, and the second follows from the property of A^{∗} described above. Combining with Corollary 3.12 and Lemma 5.3 yields homomorphisms

H^{∗}(c(K); ℚ) ⟶^{h} lim H^{∗}(B^{K}; ℚ) ⟶^{l(ϕ)} lim A_{PL}(B^{K}) ⟶^{≅} A_{PL}(c(K)),

whose composition is a quasi-isomorphism. The result follows from (5.1). □

Remark 5.6. By analogy with Remark 4.6, the proof of Theorem 5.5 extends to exponential diagrams X^{K} for which X is rationally formal; the Künneth isomorphism holds automatically, because we are working over ℚ. The product A_{PL}(X)^{⊗V} → A_{PL}(X^{V}) of the projection maps is a quasi-isomorphism, and is natural with respect to projection and inclusion of coordinates. So we may replace ϕ in (5.2) by the corresponding zig-zag of quasi-isomorphisms

H^{∗}(X^{V}; ℚ) ⟶^{≅} H^{∗}(X; ℚ)^{⊗V} ⟶^{∼} ⋯ ⟵^{∼} A_{PL}(X)^{⊗V} ⟶^{≅} A_{PL}(X^{V}),

and proceed with the remainder of the argument above.

Theorem 5.5 confirms that a minimal Sullivan model for c(K) may be constructed directly from ℚ[K]. It consists of an acyclic fibration

(5.7)  η^{K}: (S_{ℚ}(W^{K}), d) ⟶ H^{∗}(B^{K}; ℚ),

where W^{K} is an appropriately graded set of generators (necessarily exterior in odd dimensions), and provides a cofibrant replacement for H^{∗}(B^{K}; ℚ) in dgca_{ℚ}^{+}. In general, W^{K} is not easy to describe, although special cases such as Example 5.9 below are straightforward, and lead to our uniqueness result. The properties of W^{K} are linked to those of the loop space Ωc(K), whose study was begun in [21]; we expect to return to this relationship in future.

The principal calculational tool of rational homotopy theory is the Sullivan-de Rham equivalence

ho sset_{ℚ} ⟷ ho dgca_{ℚ}^{+}

of homotopy categories, which identifies homotopy classes of maps [c(K)_{0}, c(L)_{0}] with homotopy classes of morphisms [S_{ℚ}(W^{L}), S_{ℚ}(W^{K})]. Since every object of dgca_{ℚ}^{+} is fibrant, it actually suffices to consider homotopy classes of the form [S_{ℚ}(W^{L}), H^{∗}(c(K); ℚ)]; of course S_{ℚ}(W^{L}) cannot be substituted similarly, because H^{∗}(c(L); ℚ) is not usually cofibrant. Nevertheless, the function

(5.8)  [S_{ℚ}(W^{L}), H^{∗}(c(K); ℚ)] ⟶ gca_{ℚ}^{+}(H^{∗}(c(L); ℚ), H^{∗}(c(K); ℚ))

induced by taking cohomology is always surjective, and it would be of interest to understand its kernel.

Example 5.9. Let (λ(k) : 1 ≤ k ≤ t) be a sequence of disjoint subsets of V, where λ(k) has cardinality n(k), and define L to be the subcomplex of Δ(V) obtained by deleting all faces containing one or more of the λ(k). We write λ := ∪_{k}λ(k), and |λ| = n. Over any commutative ring R, the Stanley-Reisner algebra R[L] is given by quotienting out the regular sequence v_{λ(1)}, … v_{λ(t)} from S_{R}(V); so it is a complete intersection.

Over ℚ, the generating set W^{L} of (5.7) consists of V in dimension 2, and elements w(k) in dimension 2n(k) − 1, for 1 ≤ k ≤ t; the differential is given by dv_{j} = 0 for all j, and dw(k) = v_{λ(k)}. The fibration η^{L} identifies the vertices V in dimension 2, and annihilates every w(k).

Since the elements w(k) are odd dimensional, every dgca_{ℚ}^{+}-morphism S_{ℚ}(W^{L}) → H^{∗}(c(K); ℚ) is determined by its effect on V, and the function (5.8) is bijective. It follows that Aut_{ho}c(L)_{0} is isomorphic to the group of algebra automorphisms Aut ℚ[L], and is therefore a subgroup of GL(m, ℚ). It contains all matrices of the form ( \begin{matrix}M & 0 \\ N & Σ \end{matrix}), where M ∈ GL(m−n, ℚ) acts on ℚ^{V \λ}, and Σ permutes the elements of λ. The permutations act on the elements of each individual λ(k), and interchange those λ(k) which are of common cardinality. We conjecture that every automorphism of ℚ[L] may be represented by such a matrix.

It is convenient to refer to L as a complete intersection complex whenever R[L] is a complete intersection ring; a straightforward inductive argument shows that this occurs if and only if L takes the form of Example 5.9.

Our final result concerns uniqueness, and refers to Theorem 5.5 and Example 5.9. Proposition 5.10. Let X be a nilpotent CW-complex which realises a complete intersection ring ℚ[L]; then the rationalisations of X and c(L) are homotopy equivalent.

Proof. We write the generators of the cohomology algebra H^{∗}(X; ℚ) as v′_{j}, where 1 ≤ j ≤ m, and choose representing cocycles ϕ_{j} in A_{PL}(X). Each monomial v′_{M} is therefore represented by the corresponding product ϕ_{M}. It follows that ϕ_{λ(k)} is a coboundary, and there exist elements θ(k) such that dθ(k) = ϕ_{λ(k)} in A_{PL}(X) for all 1 ≤ k ≤ t.

We define a dgca_{ℚ}^{+}-morphism η: S_{ℚ}(W^{L}) → A_{PL}(X) by η(v_{j}):=ϕ_{j} and η(w(k)):=θ(k), for 1 ≤ j ≤ m and 1 ≤ k ≤ t resepectively. So η is a weak equivalence, and is a minimal model for X. Hence X and c(L) have isomorphic minimal models, and the result follows. □

Proposition 5.10 may be summarised by stating that c(L) and ℚ[L] are intrinsically formal for any complete intersection complex L. In particular, it implies that X is also formal.

References

[1] A K Bousfield and Victor K A M Gugenheim. On PL De Rham theory and rational homotopy type, volume 179 of Memoirs of the American Mathematical Society. American Mathematical Society, 1976.

[2] A K Bousfield and Daniel M Kan. Homotopy Limits, Completions and Localizations, volume 304 of Lecture notes in Mathematics. Springer Verlag, 1972.

[3] Winfried Bruns and Hans Jørgen Herzog. Cohen-Macaulay rings. Number 39 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, second edition, 1998.

[4] Victor M Buchstaber and Taras E Panov. Torus Actions and Their Applications in Topology and Combinatorics, volume 24 of University Lecture Series. American Mathematical Society, 2002.

[5] Henri Cartan and Samuel Eilenberg. Homological Algebra, volume 19 of Princeton Mathematical Series. Princeton University Press, 1956.

[6] Wojciech Chachólski and Jérôme Scherer. Homotopy theory of diagrams, volume 155 of Memoirs of the American Mathematical Society. American Mathematical Society, 2002.

[7] Michael W Davis and Tadeusz Januszkiewicz. Convex polytopes, Coxeter orbifolds and torus actions. Duke Mathematical Journal, 62:417–451, 1991.

[8] William G Dwyer. Classifying spaces and homology decompositions. In Homotopy Theoretic Methods in Group Cohomology, Advanced Courses in Mathematics CRM Barcelona, pages 1–53. Birkhäuser, 2001.

[9] William G Dwyer and J Spaliński. Homotopy theories and model categories. In Ioan M James, editor, Handbook of Algebraic Topology, pages 73–126. Elsevier, 1995.

[10] Yves Félix, Stephen Halperin, and Jean-Claude Thomas. Rational Homotopy Theory, volume 205 of Graduate Texts in Mathematics. Springer Verlag, 2001.

[11] Matthias Franz. On the integral cohomology of smooth toric varieties. Preprint, http://arxiv.org/list/math.AT/0308253, 2003.

[12] Peter Gabriel and Michel Zisman. Calculus of Fractions and Homotopy Theory. Ergebnisse der Mathematik. Springer Verlag, 1967.

[13] Jens Hollender and Rainer M Vogt. Modules of topological spaces, applications to homotopy limits and E_{∞} structures. Archiv der Mathematik, 59:115–129, 1992.

[14] Mark Hovey. Model Categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, 1999.

[15] Stefan Jackowski and James McClure. Homotopy decompositions of classifying spaces via elementary abelian subgroups. Topology, 31:113–132, 1992.

[16] Saunders MacLane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer Verlag, 1971.

[17] Michael Mandell. Cochains and homotopy type. Preprint, http://arxiv.org/list/math.AT/0311016, 2003.

[18] J Peter May. Simplicial Objects in Algebraic Topology, volume 11 of Van Nostrand Mathematical Studies. Van Nostrand Reinhold, 1967.

[19] Dietrich Notbohm and Nigel Ray. On Davis-Januszkiewicz Homotopy Types II; Completion and Globalisation. In preparation, Universities of Leicester and Manchester, 2003.

[20] Bob Oliver. Higher limits via Steinberg representations. Communications in Algebra, 22:1381–1393, 1994.

[21] Taras Panov, Nigel Ray, and Rainer Vogt. Colimits, Stanley-Reisner algebras, and loop spaces. In Algebraic Topology: Categorical Decomposition Techniques, volume 215 of Progress in Mathematics, pages 261–291. Birkhäuser, Basel, 2003.

[22] Stefan Schwede and Brooke E Shipley. Algebras and modules in monoidal model categories. Proceedings of the London Mathematical Society, 80:491–511, 2000.

[23] Richard P Stanley. Combinatorics and Commutative Algebra, 2nd edition, volume 41 of Progress in Mathematics. Birkhäuser, Boston, 1996.

[24] Rainer M Vogt. Convenient categories of topological spaces for homotopy theory. Archiv der Mathematik, 22:545–555, 1971.

[25] Volkmar Welker, Günter M Ziegler, and Rade T Živaljević. Homotopy colimits - comparison lemmas for combinatorial applications. Journal für die reine und angewandte Mathematik, 509:117–149, 1999.

Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, England

E-mail address: dn8@mcs.le.ac.uk

Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England

E-mail address: nige@ma.man.ac.uk